Curry's paradox

Named after Haskell Curry, Curry's paradox occurs in naive set theory or naive logics.

Intuitively, Curry's paradox is: "If I'm not mistaken, Y is true", where Y can be any logical statement (black is white, up is down, Gödel exists...)

If we call that statement X, then we have that X asserts "If X is true, then Y is true."

Now assume X is true. Then we have that if X is true, then Y is true. But we are assuming X is true. So Y is true. Thus, we can prove Y is true under the assumption that X is true. Which means that if X is true, then Y is true. But that's exactly what X asserts, so X is, in fact, true. Therefore, Y is true. Since Y could have been anything, this means anything can be proven via Curry's paradox.

Note that unlike Russell's paradox, this paradox does not depend on what model of negation is used, as it is completely negation-free. Thus paraconsistent logics still need to take care.

In set theories which allow unrestricted comprehension, we can prove any logical statement Y from the set

<math>X \equiv \left\{ x | x \in x \to Y \right\}</math>

The proof proceeds:

<math> \begin{matrix}

\mbox{1.} & X \in X \iff ( X \in X \to Y ) & \mbox{definition of X} \\ \mbox{2.} & X \in X \to ( X \in X \to Y ) & \mbox{from 1} \\ \mbox{3.} & X \in X \to Y & \mbox{from 2, contraction} \\ \mbox{4.} & (X \in X \to Y) \to X \in X & \mbox{from 1} \\ \mbox{5.} & X \in X & \mbox{from 3 and 4} \\ \mbox{6.} & Y & \mbox{from 3 and 5}

\end{matrix} </math>